The long homology sequence for quasi-Banach spaces, with applications
type: Item
instance of: scholarly article
title: The long homology sequence for quasi-Banach spaces, with applications
zbMATH Open document ID: 1086.46046
author: Félix Cabello Sánchez
author: Jesús M. F. Castillo
published in: Positivity
publication date: 2005-06-13T00:00:00Z
review text: Saying that \(0\to Y\to X\to Z\to 0\) is an exact sequence of quasi-Banach spaces means that \(X\) contains a subspace isomorphic to \(Y\) whose quotient is isomorphic to \(Z\). One also says that \(X\) is a twisted sum of \(Y\) and \(Z\), or an extension of \(Z\) by \(Y\). The class of all such extensions, modulo a natural equivalence relationship, is denoted \(\text{Ext}(Z,Y)\). It is well-known to algebraists that \(\text{Ext}(Z,Y)\) carries a vector space structure. The authors show that it also admits a complete semi-quasi-norm; in general, the resulting topology need not be Hausdorff. Their major result is that if \(E\) is another quasi-Banach space, then there is an exact sequence \(0\to L(Z,E)\to L(X,E)\to L(Y,E)\to\text{Ext}(Z,E)\to \text{Ext}(X,E)\to\text{Ext}(Y,E)\). The dual result for maps from (rather than into) \(E\) is also proved. The consequences of this are numerous, so we mention just a few which concern \(K\)-spaces, i.e., quasi-Banach spaces \(X\) for which \(\text{Ext}(X,E)\) is trivial whenever \(E\) is one-dimensional. One is a simple proof of the known result that quotients of Banach \(K\)-spaces are again \(K\)-spaces. Another is that the property \(\text{Ext}(\cdot,E)=0\) is a three-space property, and hence every twisted sum of an \({\mathcal L}_2\) space and an \({\mathcal L}_\infty\) space is a \(K\)-space.
Mathematics Subject Classification ID: 46M18
Mathematics Subject Classification ID: 46A16
zbMATH DE Number: 2175434
zbMATH Keywords: quasi-Banach space
zbMATH Keywords: exact sequence
MaRDI profile type: Publication
full work available at URL: s11117-002-2465-y
OpenAlex ID: W1984724842
cites work: Q3940322
cites work: Duality and twisted sums of Banach spaces
cites work: Three-space problems in Banach space theory
cites work: Q3125202
cites work: Extensions of \(c_0\)
cites work: Locally Complemented Subspaces and ℒp-Spaces for 0 < p < 1
cites work: Q3219291
cites work: Kernels of surjections from \({\mathcal L}_1\)-spaces with an application to Sidon sets
DOI: 10.1007/S11117-002-2465-Y
description: scientific article; zbMATH DE number 2175434